ASYMPTOTIC ANALYSIS OF A SYSTEM OF ALGEBRAIC EQUATIONS ARISING IN DISLOCATION THEORY

• Published in: SIAM journal on applied mathematics 2010-01, Vol.70 (7/8), p.2729-2749
• Main Authors: HALL, CAMERON L., CHAPMAN, S. JONATHAN, OCKENDON, JOHN R.
• Format: Article
• Language: English
• Subjects: Algebra; Approximation; Asymptotic methods; Asymptotic properties; Boundary layers; Continuums; Density; Differential equations; Dislocations; Maclaurin series; Mathematical analysis; Mathematical models; Monopoles; Particle density; Singular integral equations; Spatial variability; Studies; Taylor series; Theory
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/090778444

The system of algebraic equations given by $\sum\nolimits_{j = 0,j \ne i}^n {\frac{{\operatorname{sgn} ({x_i} - {x_j})}}{{{\left| {{x_i} - {x_j}} \right|}^a}}} = 1,i = 1,2,...,n,{x_0} = 0$ , appears in dislocation theory in models of dislocation pile-ups. Specifically, the case a = 1 corresponds to the simple situation where n dislocations are piled up against a locked dislocation, while the case a = 3 corresponds to n dislocation dipoles piled up against a locked dipole. We present a general analysis of systems of this type for a > 0 and n large. In the asymptotic limit n → ∞, it becomes possible to replace the system of discrete equations with a continuum equation for the particle density. For 0 < a < 2, this takes the form of a singular integral equation, while for a > 2 it is a first-order differential equation. The critical case a = 2 requires special treatment, but, up to corrections of logarithmic order, it also leads to a differential equation. The continuum approximation is valid only for i neither too small nor too close to n. The boundary layers at either end of the pile-up are also analyzed, which requires matching between discrete and continuum approximations to the main problem.